In this work, we first study the interface instability of a fluid layer flowing down on an inclined plane under periodic oscillation having both normal and lateral components. After that, we examine the effect of an insoluble surfactant covering the free surface under normal oscillation, lateral oscillation, and both normal and lateral oscillations. The time periodic linear system, corresponding to the governing equations, is treated using the Chebyshev spectral collocation method for spatial resolution, and for temporal resolution, we use the Floquet theory. We show that the stabilizing effect of normal oscillation amplitude on the gravitational instability, reported by Woods and Lin [J. Fluid Mech. 294, 391 (1995)], is strengthened by introducing lateral oscillation, and this contributes to the complete suppression of this instability. The harmonic and subharmonic zones, initially stable in the work of Woods and Lin [J. Fluid Mech. 294, 391 (1995)], are destabilized by the lateral oscillation, and the first unstable parametric resonance becomes without threshold. Conversely, the unstable domain of the gravitational instability and the second resonance zone reported by Lin, Chen, and Woods [Phys. Fluids 8, 3247 (1996)] can be reduced by introducing normal oscillation. Finally, we show that the surfactant has a stabilizing effect that contributes to accelerate the suppression of the gravitational instability and opposes the destabilizing effect of the lateral oscillation on the first subharmonic resonance to give rise to a competition between the two effects.

We investigate the effect of horizontal quasi-periodic oscillation on the stability of two superimposed immiscible fluid layers confined in a horizontal Hele-Shaw cell. To approximate real oscillations, a quasi-periodic oscillation with two incommensurate frequencies is considered. Thus, the linear stability analysis leads to a quasi-periodic oscillator, with damping, which describes the evolution of the amplitude of the interface. Two types of quasi-periodic instabilities occur: the low-wavenumber Kelvin-Helmholtz instability and the large-wavenumber resonances. We mainly show that, for equal amplitudes of the superimposed accelerations, and for a low irrational frequency ratio, there is competition between several resonance modes allowing a very large selection of the wavenumber from lower to higher values. This is a way to control the sizes of the waves. Furthermore, increasing the frequency ratio has a stabilizing effect for both types of instability whose thresholds are found to correspond to quasi-periodic solutions using the frequency spectrum. For a ratio of the two superimposed displacement amplitudes equal to unity and less than unity, the number of resonances and competition between their modes also become significant for the intermediate values of the ratio of frequencies. The effects of other physical and geometrical parameters, such as the damping coefficient, density ratio, and heights of the two fluid layers, are also examined.

We study the linear stability of two superposed layers of viscous, immiscible fluids of different densities. The whole system is subject to horizontal quasi-periodic oscillation with two incommensurates frequencies ω 1 and ω 2 . The spectral method and Floquet's theory combined with Runge-Kutta method are used to solve numericelly the linear problem. We analyse the influence of the frequencies ratio, ω = ω2 ω1 , on the mariginal stability. The numerical solution shows that the quasi-periodic excitation has a stabilizing or a destabilizing effect on the Kelvin-Helmholtz instability as well as in the parametric resonances depending on the frequency ratio and the amplitudes ratio α = a2 a1 .